Talk:Algebraic number field
|Algebraic number field has been listed as a level-4 vital article in Mathematics. If you can improve it, please do. This article has been rated as B-Class.|
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Changed the importance to top, it is about the central object of study of one of the main fields of mathematics (algebraic number theory). Doetoe 00:08, 29 May 2007 (UTC)
Unique factorization and class number
I removed the following text from that chapter, since it does not belong there:
Dirichlet L-functions L(χ, s) are a more refined variant of ζF(s). Both types of functions encode the arithmetic behavior of F and OF. For example, Dirichlet's theorem asserts that in any arithmetic progression
- a, a + m, a + 2m, ...
with coprime a and m, there are infinitely many prime numbers. This theorem is proven by showing that Dirichlet L-function is nonzero at s = 1. More recently, a computation of values of ζF(s) at all integers s has been done using much more advanced techniques including Tamagawa measures, leading to a far-reaching conjectural generalization of the description of values of more general L-functions (Tamagawa number conjecture).
- I disagree. Next to classical zeta function the Dirichlet L-function is one of the first and oldest examples. This is certainly worth mentioning. Also, as you might have noticed, I'm just in the process of enhancing the article and will provide references for the material. I will revert your removal. Jakob.scholbach (talk) 15:25, 5 June 2009 (UTC)
- I'd like to say that Dirichlet L-functions are a more refined version of the Riemann Zeta function and Hecke L-functions are a more refined version of the Dedekind zeta-functions (and all of which are automorphic L-functions).
- As for including this in this article, I would support that if it was mentioned how these values are related to invariants of the number field. It might also make sense to include these (at least as a mention) with respect to their relation to Galois representations. I do think it belongs in the algebraic number theory article. (And in case you don't have a reference handy, section 0.1 of Diamond-Flach-Guo's 2004 article in Annales Scientifiques de l'Ecole Normale Superieure gives some history with a lot of references). Cheers. RobHar (talk) 16:42, 6 June 2009 (UTC)
- Yeah sorry that sentence was some poor grammar. I meant: I would support the inclusion of the stuff about Dirichlet L-functions if the article included some explanation of how their values at integers reflect invariants of the number field. Otherwise, it seems to a bit like overload to mention more than the value at s=1. And actually there's another correction that would make things more complicated: the values of even the Riemann zeta function are not known at all integers: only the positive even integers and the negative odd integers are related to Bloch-Kato (the "critical values" of Deligne (see Corvallis)), the zeta function at positive odd integers is not very well understood (algebraicity etc). Sorry to not correct these things myself, I'm in the middle of a move; rather overwhelmed... RobHar (talk) 19:03, 6 June 2009 (UTC)
- OK. I've watered down the statement accordingly. About including stuff or not, I'd propose to keep developing the article first. If, at some point, it becomes too long, we can decide to move certain things to sup- or subarticles (i.e., algebraic number theory and L-functions etc.). Jakob.scholbach (talk) 18:34, 7 June 2009 (UTC)
The example section of this article says "The smallest and most basic number field is the field Q of rational numbers." From which one is tempted to conclude that finite fields do not exist. 126.96.36.199 (talk) 15:23, 16 April 2010 (UTC)
Different nitpick, but related. The Gaussian field "...form the first nontrivial example of a number field." In what sense, "first?" Certainly not in any ordering of fields, but since the previous example talks about the rationals being the smallest, a reader might be confused. I suspect the author means first, historically, or perhaps simply first in terms of encounters in education, but the former seems like a statement of fact, while the latter will depend entirely on education. (People will probably encounter complex numbers first,, but perhaps not the Gaussian number field.) Thomaso (talk) 16:25, 27 June 2013 (UTC)
Power integral basis?
"Moreover, a power basis obtained this way can be turned into an integral basis: if the ..."
Are you sure about this? I thought number fields don't necessarily have power integral basis?